Full Text
UDC 517.934.92
ON A PROBLEM OF N. P. ERUGIN
ON INTEGRABILITY BY QUADRATURES OF A SYSTEM
OF ORDINARY DIFFERENTIAL EQUATIONS
A. G. ASLANYAN, V. I. BURENKOV
INTRODUCTION
We consider the system of linear differential equations
\[ \frac{dX}{dt}=X[U_1\varphi_1(t)+U_2\varphi_2(t)], \tag{1} \]
where \(U_1\) and \(U_2\) are constant square matrices of order \(n\), in general complex; \(\varphi_1(t)\) and \(\varphi_2(t)\) are continuous scalar functions of \(t\); \(X\) is the unknown matrix.
N. P. Erugin [1], proceeding from the results of I. A. Lappo-Danilevskii [2] and L. M. Shneer [3], posed the following problem: to find conditions that the matrices \(U_1\) and \(U_2\) must satisfy so that the integral matrix \(X(t)\), normalized at the point \(t=t_0\), for arbitrary continuous \(\varphi_1(t)\) and \(\varphi_2(t)\), can be represented in the form
\[ X=e^S e^{U_2\psi_2}, \tag{2} \]
where
\[ \psi_2(t)=\int_{t_0}^{t}\varphi_2(\tau)\,d\tau, \]
and the matrix \(S\) commutes with its derivative, i.e.
\[ S\cdot \frac{dS}{dt}=\frac{dS}{dt}\cdot S. \]
Substituting (2) into equation (1), we obtain a differential equation with respect to \(S\), whence
\[ S=\int_{t_0}^{t} e^{U_2\psi_2}U_1e^{-U_2\psi_2}\varphi_1\,dt, \]
i.e. in this case the solution of system (1) is found by quadratures.
In the same work N. P. Erugin found the following sufficient conditions for the solvability of the posed problem:
1) \([U_1[U_2U_1]]=0,\)
2) to each eigenvalue of the matrix \(U_1\) there corresponds one elementary divisor.
Then V. V. Morozov [4] found necessary and sufficient conditions for \(U_1\) and \(U_2\) under which the solution of system (1) has the form (2). These necessary and sufficient conditions are the following:
\[ \mathfrak{A}_1\equiv [U_1[U_2U_1]]=0, \]
\[ \mathfrak A_2 \equiv \bigl[U_1[U_2[U_2U_1]]\bigr]=0, \]
\[ \cdots \tag{3} \]
\[
\mathfrak A_k \equiv [U_1[U_2\ldots[U_2[U_2U_1]\ldots]]=0,
\]
\[
\underbrace{\hspace{2.5cm}}_{k+1\ \text{brackets}}
\]
\[ \cdots \]
Various generalizations connected with this problem were considered in the works of G. F. Fedorov [5], I. M. Salakhova and G. N. Chebotarev [6]. These questions are also treated in § 5 of the monograph by N. P. Erugin [7]. The authors [8] showed that the equations of system (3) are not independent. Namely, the equations \(\mathfrak A_{2l}=0\) \((l=1,2,\ldots)\) are consequences of the equations \(\mathfrak A_{2r+1}=0\) \((r=0,1,2,\ldots)\).
§ 1. MAIN RESULTS
In the present paper we shall prove that, in the case when the matrix \(U_2\) has distinct eigenvalues \(\lambda_1,\lambda_2,\ldots,\lambda_n\), the infinite system of equations
\[ \mathfrak A_{2r+1}=0,\qquad r=0,1,\ldots \tag{4} \]
is equivalent to its part consisting of a finite number of equations, where the number \(N\) is determined by the matrix \(U_2\). The following theorem holds.
If, for some eigenvalues, equalities of the form
\[ (\lambda_i-\lambda_j)^2=(\lambda_k-\lambda_l)^2, \tag{5} \]
hold, and if there are \(\rho\) groups of mutually equal expressions of the form \((\lambda_i-\lambda_j)^2\), with \(\tau_\varkappa+1\) elements in each group, \(\varkappa=1,\ldots,\rho\), then put \(L=\tau_1+\cdots+\tau_\rho\).
Theorem 1. If the matrix \(U_2\) has distinct eigenvalues, then system (4) is equivalent to its part consisting of a finite number of equations
\[ \mathfrak A_{2r+1}=0,\qquad r=0,1,\ldots,N-1, \tag{6} \]
where
\[ N=C_n^2-L. \tag{7} \]
Combining the results of V. V. Morozov [4], the authors [8], and Theorem 1, we obtain the following result.
Theorem 2. In order that system (1) have a solution representable in the form (2), in the case when the matrix \(U_2\) has distinct eigenvalues, it is necessary and sufficient that the matrices \(U_1\) and \(U_2\) satisfy system (6).
Further, in the case when for no \(i,j,k,l\) does relation (5) hold, i.e. \(L=0\) (the distinct nonexceptional case), we shall find the matrices \(U_1\) and \(U_2\) satisfying system (6) in explicit form.
§ 2. PLAN OF THE PROOF OF THEOREM 1
First of all we introduce the following notation:
\[ (A,B)\equiv (U_1AU_1,B)=U_1AU_1B-U_1BU_1A+BU_1AU_1-AU_1BU_1. \tag{8} \]
Note the following obvious properties:
\[ (A,\ B)=-(B,\ A),\quad (A,\ A)=0, \]
\[ (\mu A,\ B)=\mu(A,\ B),\quad \mu\text{ is a number}, \tag{9} \]
\[ (A_1+A_2,\ B)=(A_1B)+(A_2B). \]
With the aid of this notation we shall reduce system (4) to a form convenient for investigation.
Lemma 1. The following formula is valid:
\[ \mathfrak A_{2r+1}=\sum_{i=0}^{r}(-1)^i C_{2r+1}^i \bigl(U_1U_2^{2r+1-i}U_1,\ U_2^i\bigr),\quad r=0,\ 1,\ 2,\ \ldots \tag{10} \]
Proof. By induction, using the relation \(C_s^i+C_s^{i-1}=C_{s+1}^i\), it is easy to verify that
\[ [U_2[U_2\ldots[U_2U_1]\ldots]] = \sum_{i=0}^{s}(-1)^i C_s^i U_2^{s-i}U_1U_2^i . \]
\[ \underbrace{\hspace{2.4cm}}_{s\ \text{brackets}} \]
It follows from this formula that
\[ \mathfrak A_{2r+1} = \sum_{s=0}^{2r+1}(-1)^s C_{2r+1}^s \bigl(U_1U_2^{2r+1-s}U_1U_2^s - U_2^{2r+1-s}U_1U_2^sU_1\bigr) = \]
\[ = \sum_{s=0}^{r}(-1)^s C_{2r+1}^s \bigl(U_1U_2^{2r+1-s}U_1U_2^s - U_2^{2r+1-s}U_1U_2^sU_1\bigr) + \]
\[ + \sum_{s=r+1}^{2r+1}(-1)^s C_{2r+1}^s \bigl(U_1U_2^{2r+1-s}U_1U_2^s - U_2^{2r+1-s}U_1U_2^sU_1\bigr) = \]
\[ = \sum_{i=0}^{r}(-1)^i C_{2r+1}^i \bigl(U_1U_2^{2r+1-i}U_1U_2^i - U_2^{2r+1-i}U_1U_2^iU_1\bigr) + \]
\[ + \sum_{i=0}^{r}(-1)^i C_{2r+1}^i \bigl(U_1U_2^iU_1U_2^{2r+1-i} - U_2^iU_1U_2^{2r+1-i}U_1\bigr)^{*} = \]
\[ = \sum_{i=0}^{r}(-1)^i C_{2r+1}^i \bigl(U_1U_2^{2r+1-i}U_1,\ U_2^i\bigr). \]
In the new notation, system (4) is written in the form
\[ \sum_{i=0}^{r}(-1)^i C_{2r+1}^i \bigl(U_2^{2r+1-i},\ U_2^i\bigr)=0,\quad r=0,\ 1,\ 2,\ \ldots \tag{11} \]
Let the matrix \(U_2\) have distinct eigenvalues \(\lambda_1,\ldots,\lambda_n\). Then (see, for example, [9], Ch. 5)
\[ f(U_2)=\sum_{i=1}^{n} f(\lambda_i)Z_i, \]
\(^*\) In the second summand the summation index has been replaced by \(s=2r+1-i\).
where \(Z_i\) are root matrices. Using this representation for \(f(U_2)=U_2^p\), we obtain from the infinite system of equations (11), with respect to an infinite number of brackets of the form \((U_2^p,U_2^q)\), an infinite linear homogeneous system of equations already with respect to a finite number of brackets of the form \((Z_i,Z_j)\). It is possible to compute effectively the coefficient matrix of this system and to find its rank. The resulting value of the rank is precisely the number \(N\) that occurs in the theorem. It turns out that \(N\) depends essentially on whether, for some \(i,j,k,l\), relation (5) is satisfied or not.
The simplest case is the multiple nonsingular case, when for no \(i,j,k,l\) does equality (5) hold. In this case \(N=C_n^2\), and the system of equations (6) is equivalent to the system of equations
\[ (Z_i,Z_j)=0,\quad i,j=1,\ldots,n,\quad i<j. \tag{12} \]
In the multiple singular case \((L\ne0)\), the system of equations (6) is also equivalent to a linear homogeneous system of equations with respect to the brackets \((Z_i,Z_j)\), but it has a more complicated form than system (12). The individual equations of this system may contain not one bracket, but a linear combination of them.
§ 3. PROOF OF THEOREM 1
Let the matrix \(U_2\) have multiple eigenvalues \(\lambda_1,\ldots,\lambda_n\). Then
\[ U_2^p=\lambda_1^p Z_1+\cdots+\lambda_n^p Z_n,\quad p=0,1,2,\ldots . \]
Starting from this representation for the powers of the matrix, we can express the bracket \((U_2^p,U_2^q)\) as a linear combination of the brackets \((Z_i,Z_j)\), i.e.
\[ (U_2^p,U_2^q)=\sum_{1\le i<j\le n} a_{ij}(Z_i,Z_j). \tag{13} \]
In view of property \((9_1)\), the summation here is taken over \(i<j\). Substituting (13) into formula (10), we obtain that
\[ \mathfrak{A}_{2r+1}=\sum_{1\le i<j\le n} A_{ij}^{(r)}(Z_i,Z_j). \]
We shall find the coefficients \(A_{ij}^{(r)}\) explicitly. The following holds.
Lemma 2.
\[ A_{ij}^{(r)}=(\lambda_i-\lambda_j)^{2r+1},\quad i,j=1,\ldots,n,\quad i<j,\quad r=0,1,2,\ldots . \]
Proof. Taking into account the properties (9), we obtain
\[ (U_2^p,U_2^q)= \left(\sum_{i=1}^{n}\lambda_i^p Z_i,\ \sum_{j=1}^{n}\lambda_j^q Z_j\right) = \sum_{i,j=1}^{n}\lambda_i^p\lambda_j^q (Z_i,Z_j) = \]
\[ = \sum_{1\le i<j\le n}\left(\lambda_i^p\lambda_j^q-\lambda_i^q\lambda_j^p\right)(Z_i,Z_j). \]
Using formula (10), we obtain,
\[ \mathfrak{A}_{2r+1}=\sum_{s=0}^{r}(-1)^s C_{2r+1}^{s} \sum_{1\leq i<j\leq n} \left(\lambda_i^{2r+1-s}\lambda_j^s-\lambda_i^s\lambda_j^{2r+1-s}\right)(Z_i,Z_j)= \]
\[ =\sum_{1\leq i<j\leq n} A_{ij}^{(r)}(Z_i,Z_j), \]
where
\[ A_{ij}^{(r)}=\sum_{s=0}^{r}(-1)^s C_{2r+1}^{s} \left(\lambda_i^{2r+1-s}\lambda_j^s-\lambda_i^s\lambda_j^{2r+1-s}\right)= \]
\[ =\sum_{s=0}^{r}(-1)^s C_{2r+1}^{s}\lambda_i^{2r+1-s}\lambda_j^s -\sum_{s=r+1}^{2r+1}(-1)^{s-1}C_{2r+1}^{s}\lambda_i^{2r+1-s}\lambda_j^{s*} = \]
\[ =\sum_{s=0}^{2r+1}(-1)^s C_{2r+1}^{s}\lambda_i^{2r+1-s}\lambda_j^s =(\lambda_i-\lambda_j)^{2r+1}. \]
Thus, system (11) is written in the form
\[ \sum_{1\leq i<j\leq n}(\lambda_i-\lambda_j)^{2r+1}(Z_i,Z_j)=0, \qquad r=0,1,2,\ldots \tag{14} \]
With respect to the elements \((Z_i,Z_j)\), whose number is finite, \(C_n^2\), we have obtained an infinite homogeneous linear system of equations.
If the equality (5) is not satisfied for any \(i,j,k,l\), then let us consider the first \(C_n^2\) equations of system (14). The determinant \(W_n\) of the resulting system of \(C_n^2\) equations with \(C_n^2\) unknowns is, up to a factor
\[
\prod_{1\leq i<j\leq n}(\lambda_i-\lambda_j)
\]
the Vandermonde determinant with respect to the elements \((\lambda_i-\lambda_j)^2\), i.e.,
\[
W_n=
\prod_{1\leq i<j\leq n}(\lambda_i-\lambda_j)
\prod\left[(\lambda_i-\lambda_j)^2-(\lambda_k-\lambda_l)^2\right]\ne 0,
\]
where the product \(\prod\) is taken over all possible nonrepeating ordered differences of the form
\[
\left[(\lambda_i-\lambda_j)^2-(\lambda_k-\lambda_l)^2\right].
\]
Consequently, this system has only the trivial solution
\[
(Z_i,Z_j)=0,\qquad i,j=1,\ldots,n,\quad i<j.
\tag{12'}
\]
It is clear that the solution \((12')\) satisfies the entire infinite system (4) and, therefore, is the unique solution of this system. Hence the infinite system (4) is equivalent to its part consisting of the first \(N=C_n^2\) equations.
Let now equalities of the form (5) be satisfied for some eigenvalues, and suppose there are \(\rho\) groups of mutually equal expressions of the form \((\lambda_i-\lambda_j)^2\), with \(\tau_\varkappa+1\) in each group, \(\varkappa=1,\ldots,\rho\). Denote the set of pairs of indices \((i,j)\) for which \((\lambda_i-\lambda_j)^2\) belongs to the first group by \(K_1\), the second group by \(K_2\), and so on. For the brackets \((Z_i,Z_j)\), where \((i,j)\in K_\varkappa\), the columns of the coefficients \((\lambda_i-\lambda_j)^{2r+1}\) coincide up to sign. Introduce new variables
\[ \text{*) In the second term the index of summation was replaced by } 2r+1-s. \]
\[ Q_\kappa=\sum_{(i,j)\in K_\kappa}(\pm)(Z_i,Z_j),\qquad \kappa=1,\ldots,\rho, \]
then with respect to these variables and the variables \((Z_i,Z_j)\), where \((i,j)\in K=K_1+\cdots+K_\rho\), we obtain an infinite system of equations. Taking into account that in this system there are \(C_n^2-L\) unknowns, where \(L=\tau_1+\cdots+\tau_\rho\), we take the first \(C_n^2-L\) equations. As above, the determinant of the resulting system is nonzero and the system has only the trivial solution
\[ \left\{ \begin{aligned} &(Z_i,Z_j)=0,\qquad i<j,\quad (i,j)\notin K,\\ &\sum_{(i,j)\in K_\kappa}(\pm)(Z_i,Z_j)=0,\qquad \kappa=1,\ldots,\rho. \end{aligned} \right. \tag{15} \]
This solution satisfies the entire infinite system (4) and, consequently, is the unique solution of this system. Therefore, in this case the infinite system (4) is equivalent to its part consisting of the first
\(N=C_n^2-L\) equations.
§ 4. SOLUTION OF N. P. ERUGIN’S PROBLEM IN THE NONMULTIPLE NONSINGULAR CASE
Let a constant matrix \(T\) (\(\det T\ne 0\)) transform the matrix \(U_2\) to diagonal form \(\widetilde U_2\), i.e. \(\widetilde U_2=TU_2T^{-1}\). In equation (1) make the substitution \(X=T^{-1}YT\); then \(Y(t_0)=E\) and
\[ \frac{dY}{dt}=Y\,[TU_1T^{-1}\varphi_1(t)+\widetilde U_2\varphi_2(t)]. \tag{1'} \]
Formula (2) assumes the form
\[ Y=Te^ST^{-1}Te^{U_2\psi_2}T^{-1}=e^{\widetilde S}e^{\widetilde U_2\psi_2}, \]
where
\[ \widetilde S=TST^{-1}\quad\text{and}\quad \widetilde S\cdot\frac{d\widetilde S}{dt}=\frac{d\widetilde S}{dt}\cdot\widetilde S. \]
Theorem 2 is applicable to equation \((1')\).
Therefore, without loss of generality we may assume in what follows that in equation (1) the matrix \(U_2\) has diagonal form. In this case
\[ Z_i= \left\| \begin{array}{ccc} 0&\vdots&0\\ \cdots&1&\cdots\\ 0&\vdots&0 \end{array} \right\|\, i . \]
It is required to solve the system
\[ (Z_i,Z_j)=0,\qquad i<j,\quad i,j=1,\ldots,n. \tag{12'} \]
Denote the elements of the matrix \(U_1\) by \(v_{ij}\) and expand the expression \((Z_i,Z_j)\), using formula (8):
\[ (Z_i, Z_j)= \left\| \begin{array}{ccccccccc} & & i & & & j & & & \\ & & \vdots & & & \vdots & & & \\ & & -v_{1j}v_{ji} & & & v_{1i}v_{ij} & & & \\ & 0 & \vdots & 0 & & \vdots & & 0 & \\ i\ \ldots & -v_{ij}v_{j1} & \ldots & -2v_{ij}v_{ji} & \ldots & v_{ij}(v_{ii}-v_{jj}) & \ldots & -v_{ij}v_{jn} & \\ & 0 & \vdots & 0 & & \vdots & & 0 & \\ j\ \ldots & v_{ji}v_{i1} & \ldots & v_{ji}(v_{ii}-v_{jj}) & \ldots & 2v_{ji}v_{ij} & \ldots & v_{ji}v_{in} & \\ & 0 & \vdots & 0 & & \vdots & & 0 & \\ & & -v_{nj}v_{ji} & & & v_{ni}v_{ij} & & & \end{array} \right\|. \tag{16} \]
Consider one equation
\[ (Z_\alpha, Z_\beta)=0. \tag{17} \]
From formula (16) it is clear that it has three independent classes of solutions:
1) \(v_{\alpha\beta}=0,\quad v_{\beta\alpha}=0\), the remaining elements arbitrary;
2) \(v_{\alpha\beta}=0,\quad v_{\alpha k}=0,\quad v_{l\beta}=0,\quad v_{\alpha\alpha}=v_{\beta\beta},\quad k,l=1,\ldots,n,\quad k,l\ne \alpha,\beta\), the remaining elements arbitrary;
3) \(v_{\beta\alpha}=0,\quad v_{\beta k}=0,\quad v_{l\alpha}=0,\quad v_{\alpha\alpha}=v_{\beta\beta},\quad k,l=1,\ldots,n,\quad k,l\ne \alpha,\beta\), the remaining elements arbitrary. Note that all matrices of class 3 are obtained by transposing the matrices of class 2.
Consider the space \(M\) of square matrices \(a\) of order \(n\):
\[ a=\|a_{ij}\|_{i,j=1}^{n}, \]
where \(a_{ij}\) are complex numbers. Introduce two operators:
\(A_{ij}^{(1)}\) is the operator of annihilating the elements \(a_{ij}\) and \(a_{ji}\);
\(A_{ij}^{(2)}\) is the operator defined as follows: \(A_{ij}^{(2)}a=\tilde a\), where the matrix \(\tilde a\) has the form
\[ \tilde a_{kl}= \begin{cases} 0, & k=i,\ l\ne i,\\ a_{ii}, & k=i,\ l=i,\\ 0, & k\ne j,\ l=j,\\ a_{ii}, & k=j,\ l=j,\\ a_{kj}, & k\ne i,\ l\ne j. \end{cases} \]
Using the operators \(A_{ij}^{(1)}\) and \(A_{ij}^{(2)}\), one can write the three classes of solutions of equation (17) in the form
\[ \begin{aligned} &1)\quad U_1=A_{\alpha\beta}^{(1)}a,\\ &2)\quad U_1=A_{\alpha\beta}^{(2)}a,\\ &3)\quad U_1=A_{\beta\alpha}^{(2)}a, \end{aligned} \]
where \(a\) is an arbitrary element of \(M\).
The operators \(A_{ij}^{(1)}\) and \(A_{ij}^{(2)}\) do not take us out of the space \(M\); therefore the product of these operators and raising them to powers are defined. The operators \(A_{ij}^{(1)}\) and \(A_{ij}^{(2)}\) possess the following properties, which follow from their definition:
-
\(A_{ij}^{(1)},\ A_{ij}^{(2)}\) are projection operators, i.e.
\[ \left[A_{ij}^{(1)}\right]^2=A_{ij}^{(1)}\quad \text{and}\quad \left[A_{ij}^{(2)}\right]^2=A_{ij}^{(2)}. \] -
The operators \(A_{ij}^{(\tau)}\) and \(A_{i'j'}^{(\tau')}\) commute.
Let us order the system \((12'')\), writing it as follows:
\[
\left\{
\begin{aligned}
X_1\equiv (Z_1,Z_2)&=0,\\
X_2\equiv (Z_1,Z_3)&=0,\\
&\ldots\ldots\ldots\\
X_{n-1}\equiv (Z_1,Z_n)&=0,\\
&\ldots\ldots\ldots\\
X_N\equiv (Z_{n-1},Z_n)&=0.
\end{aligned}
\right.
\tag{12''}
\]
Thus, \(X_m=(Z_{i(m)},Z_{j(m)})\), where \(i(m)\) and \(j(m)\) are single-valued functions of \(m\), and, conversely, \(m(i,j)\) is a single-valued function of the pair \((i,j)\). Denote
\[
A_m^{(1)}=A_{i(m)j(m)}^{(1)},
\]
\[
A_m^{(2)}=A_{i(m)j(m)}^{(2)},
\]
\[
A_m^{(3)}=A_{j(m)i(m)}^{(2)}.
\]
Let \(a\in M\). Then \(A_1^{(1)}a,\ A_1^{(2)}a,\ A_1^{(3)}(a)\) are solutions of the equation \((Z_1,Z_2)=0\):
\[
A_2^{(1)}A_1^{(1)}a,\quad A_2^{(1)}A_1^{(2)}a,\quad A_2^{(1)}A_1^{(3)}a,
\]
\[
A_2^{(2)}A_1^{(1)}a,\quad A_2^{(2)}A_1^{(2)}a,\quad A_2^{(2)}A_1^{(3)}a,
\]
\[
A_2^{(3)}A_1^{(1)}a,\quad A_2^{(3)}A_1^{(2)}a,\quad A_2^{(3)}A_1^{(3)}a
\]
give all the solutions of the system
\[
\left\{
\begin{aligned}
X_1&=0,\\
X_2&=0.
\end{aligned}
\right.
\]
And, in general, the solution of the entire system \((12'')\) is written as \(3^N=3^{C_n^2}\) solutions of the form
\[
U_1=A_N^{(i_N)}A_{N-1}^{(i_{N-1})}\ldots A_2^{(i_2)}A_1^{(i_1)}a,
\tag{18}
\]
where \(i_1,\ldots,i_N\) independently take the values \(1,2,3\).
We note that some of the solutions may coincide. Using formula (18), let us write out the solutions for \(n=2\) and \(n=3\).
-
\(n=2\),
\[ U_2= \left\| \begin{matrix} \lambda_1&0\\ 0&\lambda_2 \end{matrix} \right\|, \]
\[ U_1= \left\| \begin{matrix} v_{11}&0\\ 0&v_{22} \end{matrix} \right\|, \quad \left\| \begin{matrix} v_{11}&v_{12}\\ 0&v_{11} \end{matrix} \right\|, \quad \left\| \begin{matrix} v_{11}&0\\ v_{21}&v_{11} \end{matrix} \right\|. \] -
\(n=3\),
\[ U_2= \left\| \begin{matrix} \lambda_1&0&0\\ 0&\lambda_2&0\\ 0&0&\lambda_3 \end{matrix} \right\|, \]
\[
U_1=
\left\| \begin{matrix}
v_{11}&0&0\\
0&v_{22}&0\\
0&0&v_{33}
\end{matrix}\right\|,\quad
\left\| \begin{matrix}
v_{11}&0&0\\
v_{21}&v_{11}&0\\
0&0&v_{33}
\end{matrix}\right\|,\quad
\left\| \begin{matrix}
v_{11}&v_{12}&0\\
0&v_{11}&0\\
0&0&v_{33}
\end{matrix}\right\|,
\]
\[
\left\| \begin{matrix}
v_{11}&0&0\\
0&v_{22}&0\\
v_{31}&0&v_{11}
\end{matrix}\right\|,\quad
\left\| \begin{matrix}
v_{11}&0&v_{13}\\
0&v_{22}&0\\
0&0&v_{11}
\end{matrix}\right\|,\quad
\left\| \begin{matrix}
v_{11}&0&0\\
0&v_{22}&0\\
0&v_{32}&v_{22}
\end{matrix}\right\|,
\]
\[
\left\| \begin{matrix}
v_{11}&0&0\\
0&v_{22}&v_{23}\\
0&0&v_{22}
\end{matrix}\right\|,\quad
\left\| \begin{matrix}
v_{11}&0&0\\
v_{21}&v_{11}&0\\
v_{31}&0&v_{11}
\end{matrix}\right\|,\quad
\left\| \begin{matrix}
v_{11}&v_{12}&v_{13}\\
0&v_{11}&0\\
0&0&v_{11}
\end{matrix}\right\|,
\]
\[
\left\| \begin{matrix}
v_{11}&0&0\\
0&v_{11}&0\\
v_{31}&v_{32}&v_{11}
\end{matrix}\right\|,\quad
\left\| \begin{matrix}
v_{11}&0&v_{13}\\
0&v_{11}&v_{23}\\
0&0&v_{11}
\end{matrix}\right\|,\quad
\left\| \begin{matrix}
v_{11}&v_{12}&0\\
0&v_{11}&0\\
v_{31}&0&v_{11}
\end{matrix}\right\|,
\]
\[
\left\| \begin{matrix}
v_{11}&0&v_{13}\\
v_{21}&v_{11}&0\\
0&0&v_{11}
\end{matrix}\right\|,\quad
\left\| \begin{matrix}
v_{11}&v_{12}&0\\
0&v_{11}&0\\
0&v_{32}&v_{11}
\end{matrix}\right\|,\quad
\left\| \begin{matrix}
v_{11}&0&0\\
v_{21}&v_{11}&v_{23}\\
0&0&v_{11}
\end{matrix}\right\|.
\]
If \(n=3\), then a multiple singular case is possible, for example when
\[
\lambda_2=\frac{\lambda_1+\lambda_3}{2}.
\]
In this case, in addition to the solutions written out, the solutions of system (15) will be
\[ U_1= \left\| \begin{matrix} v_{11}&0&0\\ v_{21}&v_{11}&0\\ v_{31}&v_{32}&v_{11} \end{matrix}\right\|,\quad \left\| \begin{matrix} v_{11}&v_{12}&v_{13}\\ 0&v_{11}&v_{23}\\ 0&0&v_{11} \end{matrix}\right\|. \]
References
- N. P. Erugin, Izv. AN SSSR, Matemat., 5, 377—380, 1941.
- I. A. Lappo-Danilevskii, Trudy Fiz.-Matem. in-ta AN SSSR, vol. VIII, 1936.
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- F. R. Gantmakher, Theory of Matrices. Moscow, GITTL, 1953.
Received by the editors
December 6, 1965
Moscow Institute of Physics and Technology